Barack Obama put on his statesman’s hat as he strode into the East Room of the White House last week to implore House Republicans to enact amnesty for millions of illegal aliens and massively increase the flow of new immigrants and guest workers into the United States.

The president wanted everyone to know that his call for amnesty has nothing to do with politics. Nothing. “I’m not running for office again. I just believe this is the right thing to do,” Mr. Obama stated. But even if it did have something to do with politics, the president hastily added that doing so would be politically beneficial to Republicans.

President Obama is not the only one offering counsel to Republicans. Ezra Klein, a Democratic leaning blogger for the Washington Post, snarkily described Obama’s push for an amnesty bill as a “devious plot to destroy the Republican Party by increasing its vote share among Hispanic,” adding that “most everyone agrees [passing such a bill] would be good for the Republican Party.”

How could a party reeling from the fallout over the budget impasse resist such altruistic advice from the leader of the other party and his allies? Well, the ancient admonition, “Beware of Greeks bearing gifts,” should come to mind, finds a new analysis by the Federation for American Immigration Reform (FAIR).

For one thing, the increased “share” of the Hispanic vote Klein suggests (without evidence) the Republicans would realize from supporting an amnesty bill might actually increase their aggregate vote deficit. About 11 million Hispanics voted in the 2012 elections (out of more than 20 million who were eligible), meaning that Mitt Romney’s 27 percent share was 4.8 million votes fewer than what President Obama received. At 20 million Hispanic voters, some future Republican candidate would need to increase his/her share of those votes to 38 percent, just to maintain the same 4.8 million vote deficit.

Only one Republican presidential candidate, George W. Bush in 2004, has reached that level of Hispanic voter support. Not even his father, who was elected president just two years after major amnesty legislation was enacted by a Republican-led Senate and signed by Ronald Reagan, came anywhere close. Running against Michael Dukakis, arguably one of the weakest presidential candidates the Democrats have ever nominated, George H.W. Bush polled just 30 percent of Hispanic voters.

Similarly, John McCain, who was and remains a champion of amnesty for illegal aliens, failed miserably among Hispanic voters in his 2008 presidential bid. McCain’s 31 percent tally was only marginally better than what Romney received four years later.

Favoring or opposing amnesty for illegal aliens makes virtually no difference to a candidate’s ability to win Hispanic votes. A September 2012 Fox News Latino poll of likely Hispanic voters found that only 6 percent of them rated immigration as the most important issue in determining their votes. Far greater numbers cited the economy (48%), health care (14%) and a host of other issues as being more important to them.

The problem for Republicans is that on the issues that Hispanics really do care about, they find the Democrats’ positions far more to their liking. Hispanics tend to be significantly poorer than the population as a whole and, not coincidentally, far more likely to support big government. Seventy-five percent of Hispanics in the U.S. want government to provide more services and benefits, compared with only 19 percent who want smaller government.

Those attitudes about the role of government cannot be ignored by Republicans who cast themselves as the party lower taxes and smaller government. Worse yet, first generation Hispanics, i.e. immigrants, favor bigger government by an even wider, 81%-12% margin. In other words, there is every rational reason to expect that if Republicans were to support the amnesty and immigration increases being urged by the president and others, their reward would be an even smaller slice of a much bigger pie.

The Republican Party would have a better chance of repairing its tarnished image with Hispanic voters and others by offering an alternative vision for immigration reform. Such a vision would address the core concerns of Hispanic and other voters about jobs and economic opportunity.

Republicans would be better served by demanding that President Obama enforce existing immigration laws, allowing the 20 million-plus unemployed and underemployed Americans a shot at the jobs that already exist. Instead of bending to business’s demands for more foreign labor, they would serve their own political interests by opposing immigration increases and allowing American workers a chance at upward mobility for a change.

Sometimes good policy also makes good politics. It is now up to House Republicans to seize the opportunity.